Wednesday have must-win games against Middlesbrough, Sheffield United, Cardiff and Crystal Palace to avoid dropping into League One.
It's a tough task for the Sheffield side, who failed to find a cutting edge against a Bristol side already secure in mid-table.
Wednesday produced the positive start demanded by manager Alan Irvine with Tom Soares and Marcus Tudgay hammering half-chances over the target.
Bristol responded in emphatic manner as Lee Grant pulled off a string of stops.
Midfielder Cole Skuse was foiled by the Wednesday keeper, who snatched his long-range strike on the line before tipping Paul Hartley's free-kick over the top.
Wednesday were caught out again as Danny Haynes got the better of centre back Mark Beevers, cutting inside to pick out Maynard.
His close-range strike was palmed away by Grant before the Sheffield keeper conjured up another stop after pushing Marvin Elliott's fierce strike from 20 yards around the post.
Luke Varney almost found a way through in the 47th minute after weaving past two defenders, but his low shot hit Stephen Henderson's legs as he charged off his line.
Leon Clarke lashed a shot from 20 yards inches wide of the far post before Bristol snatched the lead in the 81st minute when Maynard rounded Beevers to slot the ball beyond Grant from 12 yards.
Substitute David Clarkson almost added a second five minutes from time but his low shot cannoned off the foot of the post.